Fold-Hopf bifurcation Analysis for a Coupled FitzHugh-Nagumo Neural System with Time Delay

A FitzHugh–Nagumo (FHN) model with delayed coupling is considered. For a critical case when the corresponding characteristic equation has a single zero root and a pair of purely imaginary roots, a complete bifurcation analysis is presented by employing the center manifold reduction and the normal form method. The Fold–Hopf bifurcation diagrams are provided to illustrate the correctness of our theoretical analysis. Whether almost periodic motion and bursting behavior occur in the FHN neural system with delayed coupling depends on the time delay in the signal transmission between the neurons.

[1]  A. Bautin.  Qualitative investigation of a particular nonlinear system: PMM vol. 39, n≗ 4, 1975, pp. 633–641 , 1975 .

[2]  J. Hale.  Theory of Functional Differential Equations , 1977 .

[3]  José M. Casado,et al.  Bursting behaviour of the FitzHugh-Nagumo neuron model subject to quasi-monochromatic noise , 1998 .

[4]  Redouane Qesmi,et al.  Center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with Fold-Hopf singularity , 2006, Appl. Math. Comput..

[5]  Xiang-Ping Yan,et al.  Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays ☆ , 2008 .

[6]  Santi Chillemi,et al.  The role of synaptic coupling in a network of FHN neuron models , 2001 .

[7]  Nikola Buric,et al.  Bifurcations due to Small Time-Lag in Coupled Excitable Systems , 2005, Int. J. Bifurc. Chaos.

[8]  Guanrong Chen,et al.  Bifurcation and synchronization of synaptically coupled FHN models with time delay , 2009 .

[9]  N. Buric,et al.  Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Jian Xu,et al.  Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay , 2010 .

[11]  Nebojša Vasović,et al.  Type I vs. type II excitable systems with delayed coupling , 2005 .

[12]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

[13]  Junjie Wei,et al.  A study of singularities for magnetic bearing systems with time delays , 2008 .

[14]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[15]  Kiyoyuki Tchizawa,et al.  On an Explicit Duck Solution and Delay in the Fitzhugh–Nagumo Equation , 1997 .

[16]  J. NAGUMOt,et al.  An Active Pulse Transmission Line Simulating Nerve Axon , 2006 .

[17]  Teresa Faria,et al.  Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities , 1996 .

[18]  Jian Xu,et al.  Simple zero singularity analysis in a coupled FitzHugh-Nagumo neural system with delay , 2010, Neurocomputing.

[19]  Jacques Bélair,et al.  Bifurcations, stability, and monotonicity properties of a delayed neural network model , 1997 .